3. THEORETICAL BACKGROUND RELATED TO CLUSTER MAGNETIC FIELDS

The synchrotron emission is produced by the spiraling motion of
relativistic electrons in a magnetic field. It is therefore the
easiest and more direct way to detect magnetic fields in astrophysical
sources. The total synchrotron emission from a source provides an
estimate of the strength of magnetic fields while the degree of
polarization is an important indicator of the field uniformity and
structure.

An electron of energy
=
mec2 (where
is the
Lorentz factor) in a magnetic field
experiences a
×
force that causes
it to follow a helical path
along the field lines, emitting radiation into a cone of half-angle
-1
about its instantaneous velocity. To the
observer, the radiation is essentially a continuum with a fairly
peaked spectrum concentrated near the critical frequency

(1)

The synchrotron power emitted by a relativistic electron is

(2)

where is the pitch
angle between the electron velocity and the magnetic field direction while
c1 and c2 depend only on fundamental
physical constants

(3)

In practical units:

(4)

(5)

From Eq. 1, it is derived that electrons of
104 in
magnetic fields of B 1 G produce synchrotron
radiation in the optical domain, whereas electrons of
105 in
magnetic fields of
B 10 G radiate
in the X-rays (see
Fig. 1). Therefore at a given frequency, the
energy (or
Lorentz factor) of the emitting electrons depends directly on the
magnetic field strengths. The higher is the magnetic field strength,
the lower is the electron energy needed to produce emission at a given
frequency. In a magnetic field of about B 1 µG, a
synchrotron radiation detected for example at 100 MHz, is produced
by relativistic electrons with
5000.

For an homogeneous and isotropic population of electrons with a
power-law energy distribution, i.e. with the particle density between
and
+
d given by

(6)

the total intensity spectrum,
in regions which are optically thin to their own radiation, varies as:

(7)

where the spectral index =
( - 1) / 2.
Below the frequency where the synchrotron emitting region becomes
optically thick, the total intensity spectrum
can be described by:

(8)

The synchrotron emission radiating from a population of relativistic
electrons in a uniform magnetic field is linearly polarized. In the
optically thin case, the degree of intrinsic linear polarization, for
a homogeneous and isotropic distribution of relativistic electrons
with a power-law spectrum as in Eq. 6, is:

(9)

with the electric (polarization) vector perpendicular to
the projection of the magnetic field onto the plane of the sky.
For typical values of the particle spectral index, the intrinsic
polarization degree is ~ 75 - 80%. In the optically thick case:

(10)

and the electric vector is parallel to the projected magnetic field.

In practice, the polarization degree detected in radio sources is much
lower than expected by the above equations. A reduction in
polarization could be due to a complex magnetic field structure whose
orientation varies either with depth in the source or over the angular
size of the beam. For instance, if one describes the magnetic field
inside an optically thin source as the superposition of two components,
one uniform Bu, the other isotropic and random
Br, the
observed degree of polarization can be approximated by
[7]:

(11)

A rigorous treatment of how the degree of polarization is
affected by the magnetic field configuration is presented by Sokoloff
et al.
[8,
9].